Splitters and near-optimal derandomization

Splitters and near-optimal derandomization Shirly Zilkha Moni naor Leonard j.schulman Aravind srinivasan

Talk Plan • Backgroud, motivation • Definitions, constructions, proofs Techniques, ideas

Backgroud, motivation • The goal of this paper is to present a fairly general method for derandomization using some combinatorial objects which are called k-restriction collections. • The heart of the method are splitters which are a generalization of k-perfect hash functions family.

(n,k)-perfect hash family • Definition: a (n,k)-perfect hash family of size s consists of a sequence φ1, φ2, …, φs of functions from {1,…,n} to {1,…,k} with the property, that for any k-subset X of {1,…,n}, there exists i such that φi is injective when restricted to X.

splitters • Definition:An (n,k,l)-splitter H is a family of functions from {1,…,n} to {1,…,l} such that for all subset of {1,…,n} S, with |S|=k, there is a h in H that splits S perfectly, i.e., into equal-sized parts (h-1(j)) ∩ S, j=1,2,…,l (or as equal as possible, if l does not divide k).

Lemma 2 • There is an explicit (n,k,k2)-splitter A(n,k) of size O(k6logklogn).

Lemma 2-Proof • The idea of the Proof:There exist explicit codes of n words over an alphabet [k2] with minimal relative distance of 1-2/k2 of length L=O(k6logklogn). We can think of each character of the coding as a function [n]->[k2].

Lemma 2- the correspondence between the code and splitters • By summing the distances we get that for any subset of k words there is an index were they all differ. • This index corresponds to the good split.

Lemma 3 • For any k<=n and for all l <= n, there is an explicit family B(n,k,l) of (n,k,l)-splitters of size (l-1n)

Lemma 3-proof • For every choice of 1<=i1<i2<…<il-1<=n, define a function h: [n]->[l] by h(s)=j iff ij-1<s<=ij, for all s in [n] (taking i0=0 and il=n). • These are all possible non-decreasing functions that get all the values from 1 to l • There are (l-1n) such functions • Every subset of size k has at least one splitting non-decreasing function

Theorem 3 (iii) • Perfect hash functions: • For k <= l <k2 ,we can produce an (n,k,l)-splitter of size ekkO(logk)logn in time linear in the output size. • also,for any l<k , an (n,k)-perfect family of hash functions of cardinality ekkO(1)(logn) (l k2)(ln k )l can be constructed deterministically in time poly(n)(k/l) k/l+1 ( k/l k2)k2k/l/(k/l)!

The end

Splitters and near-optimal derandomization